cough up vs cough

cough up is not a noun while cough is a noun.

Word	Noun	Adjective	Verb	Adverb
cough up	No	No	Yes	No
cough	Yes	No	Yes	No

As verbs, cough is a hyponym of cough up; that is, cough is a word with a more specific, narrower meaning than cough up:

Other hyponyms of cough up include ptyalise, ptyalize, spew, spit, spue, spit out, splutter, sputter.

cough up (noun)	cough (noun)
	a sudden noisy expulsion of air from the lungs that clears the air passages; a common symptom of upper respiratory infection or bronchitis or pneumonia or tuberculosis

cough up (verb)	cough (verb)
discharge (phlegm or sputum) from the lungs and out of the mouth	exhale abruptly, as when one has a chest cold or congestion
give reluctantly